Design Education
9.3 Parametric Studies 1 Sensitivity Study
Analysis o f designs has long been the domain of the computational engineer, that phase being the easiest to automate. The value o f EdenLisp is that it enables experiments to be carried out that may or may not be analytically tractable Designers are always having to deal with partial solutions where it is necessary to "suck it and see" That process is easy with EdenLisp as it will ignore partial solutions that cannot be solved and present to the user those relationships that do have values. Because of that users can play around with values o f parameters in design relationships in order to get a "feel" for the way that those parameters behave Different values of variables in the domain of the designer are quickly entered and the effect observed. Manipulating EdenLisp scripts can enable sensitivity analysis to be more flexible than conventional optimisation techniques: observations may be made on the effect of incremental changes in parameters and also the effect on the constraints imposed on the optimisation
By way of example we investigate the optimisation procedures for the selection of sizes for a compression spring suggested by [Siddall, 1982]. Loading is static compression and the spring ends are assumed to be closed and ground flat.
DESIGN F.DIICA TION 160
Notation
N = Number o r active coils (usually end coils arc inactive, so N no. coils - 2)
D = Mean diam eter of the coil (mm)
11 = Diameter o f the wire used for the spring (mm): usually a preferred standard size. G = Bulk Modulus (MPa)
S = Maximum shear stress of spring material (MPa) Fmax = Maximum working Load (N)
lm ax - Maximum free length (mm) Dmax ~ Maximum coil diameter (mm)
dm m = Minimum wire diameter (mm) Fp = Preload compression force (N)
f.y
= End CociTicient of the spring (= 1 for parallel ends with one fixed, one free)Spm = Maximum allowable deflection under preload (mm)
= Deflection from preload position to maximum load position (mm)
1 Criterion Function
The optimisation criterion is that of minimising the volume o f the spring wire used. ( / = — D d 2{N + 2)
4
This criterion is subject to a series o f 8 constraints d>j as follows
2 Constraints
1. Strength. The shear stress in the spring must be less than the yield shear strength of the spring material, S,htar. The stress has two components: a shear force on the cross-section o f the wire, and torsion o f the wire. The total shear stress in the wire is expressed in terms of the loading and a spring index a function o f D J. The stress constraint <t>, is
<*>! > 0
2. Deflection constraint
The stiffness o f a coil spring K (N/mm) is
K (h i* ~ HNDi
The deflection (mm) of the spring under maximum static load is f-'max K The
spring length under load Fmax is 105% o f the solid length the spring The free length is given by
lf = <5 + 1 05(iV + 2)</ The deflection constraint is
DESIGN EDUCATION 161
3. Wire Diameter constraint
The wire diameter must not be less than the minimum specified.
^ = d - d ^ -0 4. Coil Diameter
The outside diameter of the coil must not exceed the maximum
*>>=Dm - D - d > 0 5. Coil Winding restriction
The mean coil diameter must be at least three times the wire diameter to prevent it being too tightly wound.
4>, = C - 3 SO
6. Preload deflection
The preload deflection must be less than F ^K and so the constraint is
7. Total deflection constraint
The combined deflection must be consistent with the free length o f the unloaded spring
fi. Specified deflection
The deflection from the preload position to the maximum load position must be
We do a parameter study by entering the equations as EdenLisp definitions, and
the constraints as P h il, Phi2, etc. as definitions with violation messages.
Drawings o f the spring under the conditions applied are easy to do by associating the variables with appropriate geometrical models Possible EdenLisp script and some results follow.
);; EDENLISP Exercise in Optimising compression spring dimensions tft l Definitions of relationships Vol = p i A2*D*dA2*(N+2)/4 K = G*dA4/(8*N*DA3) Lf » d w +■ 1.05* (N+2) delpm = Fp/K K - S w£0
DESIGN EDUCATION 162
; definitions of constraints
Phil = S - 8*Cf *Elnax*D/ (pi*lA3)
Phi2 = Imax - Lf Phi3 = d - dmin
Phi 4 - Dmax - D - d
Phi 5 = C - 3 Phi6 « delpm - d e l p
Phi7 « lmax - d e l p - <Rnax-Fp)/K - 1.05*d*(N+2) PhiB = (Ftaax-Fp) / K - dpm
constraintl = if Phil > = 0 then print "Phil-ve. Stress too high"
constraint2= if P h i 2 > = 0 t h e n print "Phi2-ve. Spring solid with no load" constraints = if P h i 3 > = 0 t h e n print "Phi3-ve, spring wire dia too small" constraint4= if P h i 4 > = 0 t h e n print "Phi4-ve, Coil dia exceeds design max" constraints = if P h i 5 > = 0 t h e n print "Phi5-ve. Coil dia too small"
constraint6 = if P h i 6 > = 0 t h e n print "Phi6-ve. Preload too great: spring solid"
constraint?= if P h i 7 > = 0 t h e n print "Phi7-ve. Combined deflection too great" constraints = if P h i 8 > = 0 t h e n print "Phi8-ve. Load too great: spring solid"
;;; Results of two sets of values of variables d, D, N and material
Strength: S 1100 Stiffness: K 96.02
Elastic Modulus: E 205000 Deflection: 5 46.24
Bulk Modulus: G 80000 Free L e n g t h : Lf 259.39
Force: Ftnax 4440 Delp 13.87
Length: Lmax 355 CF 1.58
Wire diameter: Dmin 5
Outer diameter: Dmax 75 Optimisation: Volume 73629.59
Preload: Fp 1332 Constraints to be >0
Preload def: dpm 150 Phil 6.28
Deflection: dw 32 Phi2 95.61
End Coefficient: C E 1 Phi 3 2
var: d 7 Phi 4 47 var: D 21 Phi 5 0 var: N 27 Phi 6 136.13 constant: pi 3.1416 Phi 7 0 Ratio D/d: C 3 Phi 8 0.37 Strength: S 676.69 Stiffness: K 97.35
Elastic Modulus: E 207000 Deflection: ft 45.61
Bulk Modulus: G 80000 Free Length: Lf 354.62
Force: Rnax 4440 Del: p 13.66
Length: Imax 355 CF 1.55
Wire diameter: Cktiin 5
Outer diameter: L*nax 75 Optimisation: Volume 182991.0
Preload: Ep 1330 Constraints to be 0
Preload def: dpm 150 Phil 2.31
Deflection: dw 35 Phi2 0.38
End Coefficient: CE 1 Phi 3 4
var: d 9 Phi 4 38
var: D 28 Phi 5 0.11
var: N 30.7 Phi 6 136.34
constant : pi 3.1416 Phi 7 0
Ratio D/d: C 3.11 Phi 8 -3.06
DESIGN EDUCATION 163
Optimisation may be carried out as a batch process using numerical methods such as Siddall suggests in his text (op cit.). The advantage of interactive study of sensitivity is that the student can observe what happens to the optimisation function as different parameters are manually varied. Changing the values of variables in EdenLisp just involves redefinition, so the designer can quickly study the behaviour o f important parameters and can just as easily change constraints. The tables show the results o f varying the material strength, wire diameter and coil outer diameter. By checking what is happening to the constraints at the same time it is possible to see whether the constraints themselves are reasonable. When the value Phi8 = - 3.06 is obtained in the second table, indicating a constraint violation, we can not only to vary the main parameter to get us out o f violation but also see whether the specified deflection from pre-load position to maximum load position 8W needs to be changed. It is difficult for a preconceived optimisation analysis to anticipate those kinds o f adjustments to constraints; and for the student to know what to do with a result o f an optimisation analysis when constraints are not examined in that way.